Given below are two statements:
Statement I : The limiting molar conductivity of KCl (strong electrolyte) is higher compared to that of CH$$_3$$COOH (weak electrolyte).
Statement II : Molar conductivity decreases with decrease in concentration of electrolyte.
In the light of the above statements, choose the most appropriate answer from the options given below:

We begin by recalling the definition of limiting molar conductivity (also called molar conductivity at infinite dilution). According to Kohlrausch’s Law of Independent Migration of Ions,

$$\Lambda_m^{\circ} = \lambda^{\circ}_{+} + \lambda^{\circ}_{-},$$

where $$\lambda^{\circ}_{+}$$ and $$\lambda^{\circ}_{-}$$ are the ionic molar conductivities of the cation and the anion at infinite dilution. These values depend only on the speed of the individual ions in a very dilute solution and are completely independent of whether the electrolyte is strong or weak.

For the strong electrolyte $$\text{KCl}$$ we have the ions $$\text{K}^{+}$$ and $$\text{Cl}^{-}$$. Their ionic molar conductivities at $$25^{\circ}\text{C}$$ are approximately

$$\lambda^{\circ}_{\text{K}^{+}} \approx 73.5\ \text{S cm}^{2}\text{mol}^{-1}, \qquad \lambda^{\circ}_{\text{Cl}^{-}} \approx 76.3\ \text{S cm}^{2}\text{mol}^{-1}.$$

Adding these, we obtain

$$\Lambda_m^{\circ}(\text{KCl}) = 73.5 + 76.3 = 149.8\ \text{S cm}^{2}\text{mol}^{-1}.$$

For the weak electrolyte $$\text{CH}_3\text{COOH}$$ the ions that eventually appear on complete (hypothetical) dissociation are $$\text{H}^{+}$$ and $$\text{CH}_3\text{COO}^{-}$$. Their ionic molar conductivities at $$25^{\circ}\text{C}$$ are approximately

$$\lambda^{\circ}_{\text{H}^{+}} \approx 349.6\ \text{S cm}^{2}\text{mol}^{-1}, \qquad \lambda^{\circ}_{\text{CH}_3\text{COO}^{-}} \approx 40.9\ \text{S cm}^{2}\text{mol}^{-1}.$$

Hence,

$$\Lambda_m^{\circ}(\text{CH}_3\text{COOH}) = 349.6 + 40.9 = 390.5\ \text{S cm}^{2}\text{mol}^{-1}.$$

Clearly, $$\Lambda_m^{\circ}(\text{CH}_3\text{COOH}) > \Lambda_m^{\circ}(\text{KCl}).$$ Therefore Statement I, which claims the opposite, is false.

Now we analyse how molar conductivity varies with concentration. The molar conductivity $$\Lambda_m$$ at any concentration $$c$$ is defined as

$$\Lambda_m = \frac{\kappa}{c},$$

where $$\kappa$$ is the measured conductivity of the solution. Experimentally we observe that on dilution (that is, when $$c$$ decreases), inter-ionic interactions become weaker, the ions move more freely, $$\kappa$$ does not drop as fast as $$c$$, and consequently $$\Lambda_m$$ actually increases. Hence molar conductivity increases with decrease in concentration for both strong and weak electrolytes.

Therefore Statement II, which says that molar conductivity decreases on dilution, is also false.

Since both statements are false, the option that asserts “Both Statement I and Statement II are false” is correct.

Hence, the correct answer is Option D.